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LETTER TO THE EDITOR
1975]
173
Now if x=0 (modpi Fx(n)=1 (mod p) for all n, by the definition of Fx(n).
lix£0(modpl
from Lemma 5 there exists a number a such that Fx(a) = O (mod p), we assume that a is the
least such number, and a > 1 since Fx(1) = 7 for a l l * . It can be shown inductively that Fx(n +a)^sFx(n)
(mod p)
for all/?, where s = Fx(a+ 1) (mod pi and s ^ 0 since s = 0 would imply Fx(a1) = 0(mod p). Then if Fx(r)= 0
(modpi there exists r' such that
r' = r(modal
0 < /*' < a,
and
Fx(r') = 0
(modpi
By the definition of a, /•'< a is absurd, therefore r'= a.
Let P be prime and p a prime factor of Fx(Pl Then
FX(P) = 0 (modp)
and x £0 (modp)
since, if x= 0 (modpi Fx(n) = 1 (modp) for all n.
Thus P = 0 (mod a) and since P is prime, P = a. Let p' be either p,p - 1, or p + 1f such that
Fx(p1
= 0 (mod p)
(from Lemma 3). Then p' is an integral multiple of P and the theorem follows.
I mentioned this result to Dr. P.M. Lee of York University and he has pointed out to me that Lemma 3 can be derived from H. Siebeck's work on recurring series (L.E. Dickson, History of the Theory of Numbers, p. 394f). A colleague of his has also discovered a non-elementary proof of the above theorem.
I am myself only an amateur mathematician, so I would ask you to excuse any resulting awkwardnesses in my presentation of this theorem and proof.
Yours faithfully,
Alexander G. Abercrombie
[Continued from Page 146.]
^kkkk^k
There is room for considerable work regarding possible lengths of periods. For various values of p and q we found
periods of lengths: 1, 2, 8, 9, 17, 25, 33, 35, 42, 43, 61, 69.
GENERALIZED PERIODS
For various sequence types, it is possible to arrive at generalized periods. Some examples are the following.
(p,p - 1): 2p -2,2p3, 2p - 3, 2p - 2, 2p, 2p +2, 2p +3, 2p +2, 2pf where/? is large enough to make all quantities positive.
fa;p): 2p, 2p +2, 2p, 2p + 1,2p- 7, 2p, 2p - 7, 2p + 7, where p>2.
2p - 1,2p + 7, 2p - 1, 2p +2, 2p, 2p + 3, 2p, 2p +2, where p>2, and many others.
(p + lp): 2p- 1,2p,2p+2,2p+4,2p+5,2p+4,2p.
+ 2,2p,2p7 f p r / ? > 3 . (Period of length 9)
2p( 2p + I 2p+5, 2p+5, 2p+5, 2p + 7, 2p, 2p ~3f2p~
1, 2p - I 2p+4, 2p +4, 2p + 7, 2p+3f
2p +2, 2p -3f2p2, 2p - 3, 2p +2, 2p+3f 2p+8t 2p + 7, 2p +4, 2p+4t 2p -1,2p7,2p - 3,
for p>24
(period of length 26), and many others.
A schematic method was used which made the work of arriving at these results somewhat less laborious.
NON-PERIODIC SEQUENCES
Sn studying the sequences (3,4), non-periodic sequences of a quasi-periodic type were found. They have the peculiar property that alternate terms form a regular pattern in groups of four, while the intermediate terms between these
pattern terms become unbounded. This situation arises in sequences (p,q) for which q is greater than p.
As an example of such a non-periodic sequence in the case (4,7) the sequence beginning with 1,3,4, follows:
1, 3, 4, 37, 59, 124, 25, 17, 2, 6, 3, 27, 22, 93, 20, 34, 3, 13, 3, 35, 13, 99, 14, 58, 4, 31, 3, 58, 9, 148, 12, 121, 4,
72, 3, 129, 8, 312, 11, 279, 4, 179, 3, 317, 8, 751, 10, 663, 4, 466, 3, 819, 8, 1922, 10, 1687, 4, 1183, 3, 2074, 8,
4850, 10,4249,4,2976,3,5211,8, 12170, 10,....
Note the regular periodicidity of 3,8,10,4 with the sets of intermediate terms increasing as the sequence progresses.
The various types of non-periodic sequence for (4,7) are:
[Continued on Page 184.]